LMFDB - Embedded newform 3328.2.a.bk.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3328,2,Mod(1,3328)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3328.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3328, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3328 = 2^{8} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3328.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,-2,0,0,0,14] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$26.5742137927$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.13448.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 7x^{2} + 2 $ x^4 - 7*x^2 + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1664)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.58874$ of defining polynomial
Character	$\chi$	$=$	3328.1

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.81616 q^{3} +2.70156 q^{5} +2.58874 q^{7} +0.298438 q^{9} +0.772577 q^{11} -1.00000 q^{13} +4.90647 q^{15} -0.701562 q^{17} +4.40490 q^{19} +4.70156 q^{21} +3.63232 q^{23} +2.29844 q^{25} -4.90647 q^{27} -2.00000 q^{29} -5.95005 q^{31} +1.40312 q^{33} +6.99364 q^{35} +6.70156 q^{37} -1.81616 q^{39} +11.4031 q^{41} +10.0839 q^{43} +0.806248 q^{45} -9.31137 q^{47} -0.298438 q^{49} -1.27415 q^{51} -1.40312 q^{53} +2.08717 q^{55} +8.00000 q^{57} +2.85974 q^{59} +0.772577 q^{63} -2.70156 q^{65} -12.6727 q^{67} +6.59688 q^{69} -1.04358 q^{71} +7.40312 q^{73} +4.17433 q^{75} +2.00000 q^{77} +2.08717 q^{79} -9.80625 q^{81} +9.58237 q^{83} -1.89531 q^{85} -3.63232 q^{87} +11.4031 q^{89} -2.58874 q^{91} -10.8062 q^{93} +11.9001 q^{95} +10.0000 q^{97} +0.230566 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{5} + 14 q^{9} - 4 q^{13} + 10 q^{17} + 6 q^{21} + 22 q^{25} - 8 q^{29} - 20 q^{33} + 14 q^{37} + 20 q^{41} - 48 q^{45} - 14 q^{49} + 20 q^{53} + 32 q^{57} + 2 q^{65} + 52 q^{69} + 4 q^{73} + 8 q^{77}+ \cdots + 40 q^{97}+O(q^{100}) $ 4 * q - 2 * q^5 + 14 * q^9 - 4 * q^13 + 10 * q^17 + 6 * q^21 + 22 * q^25 - 8 * q^29 - 20 * q^33 + 14 * q^37 + 20 * q^41 - 48 * q^45 - 14 * q^49 + 20 * q^53 + 32 * q^57 + 2 * q^65 + 52 * q^69 + 4 * q^73 + 8 * q^77 + 12 * q^81 - 46 * q^85 + 20 * q^89 + 8 * q^93 + 40 * q^97

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	1.81616	1.04856	0.524280	−	0.851546i	$-0.324334\pi$
$3$	1.81616	1.04856	0.524280	+	0.851546i	$0.324334\pi$
$4$	0	0
$5$	2.70156	1.20818	0.604088	−	0.796918i	$-0.293538\pi$
$5$	2.70156	1.20818	0.604088	+	0.796918i	$0.293538\pi$
$6$	0	0
$7$	2.58874	0.978451	0.489225	−	0.872157i	$-0.337280\pi$
$7$	2.58874	0.978451	0.489225	+	0.872157i	$0.337280\pi$
$8$	0	0
$9$	0.298438	0.0994793
$10$	0	0
$11$	0.772577	0.232941	0.116470	−	0.993194i	$-0.462842\pi$
$11$	0.772577	0.232941	0.116470	+	0.993194i	$0.462842\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	4.90647	1.26685
$16$	0	0
$17$	−0.701562	−0.170154	−0.0850769	−	0.996374i	$-0.527114\pi$
$17$	−0.701562	−0.170154	−0.0850769	+	0.996374i	$0.527114\pi$
$18$	0	0
$19$	4.40490	1.01055	0.505276	−	0.862958i	$-0.331391\pi$
$19$	4.40490	1.01055	0.505276	+	0.862958i	$0.331391\pi$
$20$	0	0
$21$	4.70156	1.02596
$22$	0	0
$23$	3.63232	0.757391	0.378696	−	0.925521i	$-0.376373\pi$
$23$	3.63232	0.757391	0.378696	+	0.925521i	$0.376373\pi$
$24$	0	0
$25$	2.29844	0.459688
$26$	0	0
$27$	−4.90647	−0.944251
$28$	0	0
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	0	0
$31$	−5.95005	−1.06866	−0.534330	−	0.845276i	$-0.679436\pi$
$31$	−5.95005	−1.06866	−0.534330	+	0.845276i	$0.679436\pi$
$32$	0	0
$33$	1.40312	0.244253
$34$	0	0
$35$	6.99364	1.18214
$36$	0	0
$37$	6.70156	1.10173	0.550865	−	0.834594i	$-0.314298\pi$
$37$	6.70156	1.10173	0.550865	+	0.834594i	$0.314298\pi$
$38$	0	0
$39$	−1.81616	−0.290818
$40$	0	0
$41$	11.4031	1.78087	0.890434	−	0.455112i	$-0.150401\pi$
$41$	11.4031	1.78087	0.890434	+	0.455112i	$0.150401\pi$
$42$	0	0
$43$	10.0839	1.53779	0.768894	−	0.639377i	$-0.220808\pi$
$43$	10.0839	1.53779	0.768894	+	0.639377i	$0.220808\pi$
$44$	0	0
$45$	0.806248	0.120188
$46$	0	0
$47$	−9.31137	−1.35820	−0.679101	−	0.734045i	$-0.737630\pi$
$47$	−9.31137	−1.35820	−0.679101	+	0.734045i	$0.737630\pi$
$48$	0	0
$49$	−0.298438	−0.0426340
$50$	0	0
$51$	−1.27415	−0.178417
$52$	0	0
$53$	−1.40312	−0.192734	−0.0963670	−	0.995346i	$-0.530722\pi$
$53$	−1.40312	−0.192734	−0.0963670	+	0.995346i	$0.530722\pi$
$54$	0	0
$55$	2.08717	0.281433
$56$	0	0
$57$	8.00000	1.05963
$58$	0	0
$59$	2.85974	0.372307	0.186153	−	0.982521i	$-0.440398\pi$
$59$	2.85974	0.372307	0.186153	+	0.982521i	$0.440398\pi$
$60$	0	0
$61$	0	0		−	1.00000i	$-0.5\pi$
$61$	0	0			1.00000i	$0.5\pi$
$62$	0	0
$63$	0.772577	0.0973356
$64$	0	0
$65$	−2.70156	−0.335088
$66$	0	0
$67$	−12.6727	−1.54821	−0.774107	−	0.633054i	$-0.781801\pi$
$67$	−12.6727	−1.54821	−0.774107	+	0.633054i	$0.781801\pi$
$68$	0	0
$69$	6.59688	0.794171
$70$	0	0
$71$	−1.04358	−0.123850	−0.0619252	−	0.998081i	$-0.519724\pi$
$71$	−1.04358	−0.123850	−0.0619252	+	0.998081i	$0.519724\pi$
$72$	0	0
$73$	7.40312	0.866470	0.433235	−	0.901281i	$-0.357372\pi$
$73$	7.40312	0.866470	0.433235	+	0.901281i	$0.357372\pi$
$74$	0	0
$75$	4.17433	0.482010
$76$	0	0
$77$	2.00000	0.227921
$78$	0	0
$79$	2.08717	0.234824	0.117412	−	0.993083i	$-0.462540\pi$
$79$	2.08717	0.234824	0.117412	+	0.993083i	$0.462540\pi$
$80$	0	0
$81$	−9.80625	−1.08958
$82$	0	0
$83$	9.58237	1.05180	0.525901	−	0.850546i	$-0.323728\pi$
$83$	9.58237	1.05180	0.525901	+	0.850546i	$0.323728\pi$
$84$	0	0
$85$	−1.89531	−0.205576
$86$	0	0
$87$	−3.63232	−0.389426
$88$	0	0
$89$	11.4031	1.20873	0.604364	−	0.796708i	$-0.293427\pi$
$89$	11.4031	1.20873	0.604364	+	0.796708i	$0.293427\pi$
$90$	0	0
$91$	−2.58874	−0.271373
$92$	0	0
$93$	−10.8062	−1.12056
$94$	0	0
$95$	11.9001	1.22093
$96$	0	0
$97$	10.0000	1.01535	0.507673	−	0.861550i	$-0.330506\pi$
$97$	10.0000	1.01535	0.507673	+	0.861550i	$0.330506\pi$
$98$	0	0
$99$	0.230566	0.0231728
$100$	0	0
$101$	−6.80625	−0.677247	−0.338624	−	0.940922i	$-0.609961\pi$
$101$	−6.80625	−0.677247	−0.338624	+	0.940922i	$0.609961\pi$
$102$	0	0
$103$	−13.9873	−1.37821	−0.689103	−	0.724663i	$-0.741995\pi$
$103$	−13.9873	−1.37821	−0.689103	+	0.724663i	$0.741995\pi$
$104$	0	0
$105$	12.7016	1.23955
$106$	0	0
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	−1.29844	−0.124368	−0.0621839	−	0.998065i	$-0.519807\pi$
$109$	−1.29844	−0.124368	−0.0621839	+	0.998065i	$0.519807\pi$
$110$	0	0
$111$	12.1711	1.15523
$112$	0	0
$113$	−1.40312	−0.131995	−0.0659974	−	0.997820i	$-0.521023\pi$
$113$	−1.40312	−0.131995	−0.0659974	+	0.997820i	$0.521023\pi$
$114$	0	0
$115$	9.81294	0.915061
$116$	0	0
$117$	−0.298438	−0.0275906
$118$	0	0
$119$	−1.81616	−0.166487
$120$	0	0
$121$	−10.4031	−0.945739
$122$	0	0
$123$	20.7099	1.86735
$124$	0	0
$125$	−7.29844	−0.652792
$126$	0	0
$127$	−10.8970	−0.966949	−0.483474	−	0.875358i	$-0.660625\pi$
$127$	−10.8970	−0.966949	−0.483474	+	0.875358i	$0.660625\pi$
$128$	0	0
$129$	18.3141	1.61246
$130$	0	0
$131$	−4.90647	−0.428680	−0.214340	−	0.976759i	$-0.568760\pi$
$131$	−4.90647	−0.428680	−0.214340	+	0.976759i	$0.568760\pi$
$132$	0	0
$133$	11.4031	0.988776
$134$	0	0
$135$	−13.2551	−1.14082
$136$	0	0
$137$	14.0000	1.19610	0.598050	−	0.801459i	$-0.295942\pi$
$137$	14.0000	1.19610	0.598050	+	0.801459i	$0.295942\pi$
$138$	0	0
$139$	−12.1711	−1.03234	−0.516170	−	0.856486i	$-0.672643\pi$
$139$	−12.1711	−1.03234	−0.516170	+	0.856486i	$0.672643\pi$
$140$	0	0
$141$	−16.9109	−1.42416
$142$	0	0
$143$	−0.772577	−0.0646062
$144$	0	0
$145$	−5.40312	−0.448705
$146$	0	0
$147$	−0.542011	−0.0447043
$148$	0	0
$149$	−12.8062	−1.04913	−0.524564	−	0.851371i	$-0.675772\pi$
$149$	−12.8062	−1.04913	−0.524564	+	0.851371i	$0.675772\pi$
$150$	0	0
$151$	23.8406	1.94012	0.970062	−	0.242856i	$-0.0780844\pi$
$151$	23.8406	1.94012	0.970062	+	0.242856i	$0.0780844\pi$
$152$	0	0
$153$	−0.209373	−0.0169268
$154$	0	0
$155$	−16.0744	−1.29113
$156$	0	0
$157$	4.59688	0.366871	0.183435	−	0.983032i	$-0.441278\pi$
$157$	4.59688	0.366871	0.183435	+	0.983032i	$0.441278\pi$
$158$	0	0
$159$	−2.54830	−0.202093
$160$	0	0
$161$	9.40312	0.741070
$162$	0	0
$163$	−16.8470	−1.31956	−0.659780	−	0.751459i	$-0.729351\pi$
$163$	−16.8470	−1.31956	−0.659780	+	0.751459i	$0.729351\pi$
$164$	0	0
$165$	3.79063	0.295100
$166$	0	0
$167$	−14.7598	−1.14215	−0.571076	−	0.820897i	$-0.693474\pi$
$167$	−14.7598	−1.14215	−0.571076	+	0.820897i	$0.693474\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	1.31459	0.100529
$172$	0	0
$173$	−21.4031	−1.62725	−0.813625	−	0.581390i	$-0.802509\pi$
$173$	−21.4031	−1.62725	−0.813625	+	0.581390i	$0.802509\pi$
$174$	0	0
$175$	5.95005	0.449782
$176$	0	0
$177$	5.19375	0.390386
$178$	0	0
$179$	−24.0712	−1.79917	−0.899584	−	0.436749i	$-0.856130\pi$
$179$	−24.0712	−1.79917	−0.899584	+	0.436749i	$0.856130\pi$
$180$	0	0
$181$	−14.0000	−1.04061	−0.520306	−	0.853980i	$-0.674182\pi$
$181$	−14.0000	−1.04061	−0.520306	+	0.853980i	$0.674182\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	18.1047	1.33108
$186$	0	0
$187$	−0.542011	−0.0396358
$188$	0	0
$189$	−12.7016	−0.923903
$190$	0	0
$191$	−5.17748	−0.374629	−0.187314	−	0.982300i	$-0.559978\pi$
$191$	−5.17748	−0.374629	−0.187314	+	0.982300i	$0.559978\pi$
$192$	0	0
$193$	20.8062	1.49767	0.748833	−	0.662758i	$-0.230614\pi$
$193$	20.8062	1.49767	0.748833	+	0.662758i	$0.230614\pi$
$194$	0	0
$195$	−4.90647	−0.351360
$196$	0	0
$197$	0.104686	0.00745859	0.00372930	−	0.999993i	$-0.498813\pi$
$197$	0.104686	0.00745859	0.00372930	+	0.999993i	$0.498813\pi$
$198$	0	0
$199$	−6.72263	−0.476555	−0.238277	−	0.971197i	$-0.576583\pi$
$199$	−6.72263	−0.476555	−0.238277	+	0.971197i	$0.576583\pi$
$200$	0	0
$201$	−23.0156	−1.62340
$202$	0	0
$203$	−5.17748	−0.363388
$204$	0	0
$205$	30.8062	2.15160
$206$	0	0
$207$	1.08402	0.0753447
$208$	0	0
$209$	3.40312	0.235399
$210$	0	0
$211$	8.53879	0.587835	0.293917	−	0.955831i	$-0.405041\pi$
$211$	8.53879	0.587835	0.293917	+	0.955831i	$0.405041\pi$
$212$	0	0
$213$	−1.89531	−0.129865
$214$	0	0
$215$	27.2424	1.85792
$216$	0	0
$217$	−15.4031	−1.04563
$218$	0	0
$219$	13.4453	0.908546
$220$	0	0
$221$	0.701562	0.0471922
$222$	0	0
$223$	−20.2083	−1.35325	−0.676625	−	0.736328i	$-0.736558\pi$
$223$	−20.2083	−1.35325	−0.676625	+	0.736328i	$0.736558\pi$
$224$	0	0
$225$	0.685941	0.0457294
$226$	0	0
$227$	0.230566	0.0153032	0.00765161	−	0.999971i	$-0.497564\pi$
$227$	0.230566	0.0153032	0.00765161	+	0.999971i	$0.497564\pi$
$228$	0	0
$229$	22.9109	1.51400	0.756999	−	0.653417i	$-0.226665\pi$
$229$	22.9109	1.51400	0.756999	+	0.653417i	$0.226665\pi$
$230$	0	0
$231$	3.63232	0.238989
$232$	0	0
$233$	21.5078	1.40902	0.704512	−	0.709692i	$-0.251166\pi$
$233$	21.5078	1.40902	0.704512	+	0.709692i	$0.251166\pi$
$234$	0	0
$235$	−25.1552	−1.64095
$236$	0	0
$237$	3.79063	0.246228
$238$	0	0
$239$	27.9341	1.80691	0.903453	−	0.428686i	$-0.141023\pi$
$239$	27.9341	1.80691	0.903453	+	0.428686i	$0.141023\pi$
$240$	0	0
$241$	15.4031	0.992202	0.496101	−	0.868265i	$-0.334764\pi$
$241$	15.4031	0.992202	0.496101	+	0.868265i	$0.334764\pi$
$242$	0	0
$243$	−3.09031	−0.198243
$244$	0	0
$245$	−0.806248	−0.0515093
$246$	0	0
$247$	−4.40490	−0.280277
$248$	0	0
$249$	17.4031	1.10288
$250$	0	0
$251$	−26.4294	−1.66821	−0.834104	−	0.551607i	$-0.814015\pi$
$251$	−26.4294	−1.66821	−0.834104	+	0.551607i	$0.814015\pi$
$252$	0	0
$253$	2.80625	0.176427
$254$	0	0
$255$	−3.44219	−0.215559
$256$	0	0
$257$	10.7016	0.667545	0.333773	−	0.942654i	$-0.391678\pi$
$257$	10.7016	0.667545	0.333773	+	0.942654i	$0.391678\pi$
$258$	0	0
$259$	17.3486	1.07799
$260$	0	0
$261$	−0.596876	−0.0369457
$262$	0	0
$263$	19.1647	1.18175	0.590874	−	0.806764i	$-0.298783\pi$
$263$	19.1647	1.18175	0.590874	+	0.806764i	$0.298783\pi$
$264$	0	0
$265$	−3.79063	−0.232856
$266$	0	0
$267$	20.7099	1.26743
$268$	0	0
$269$	27.4031	1.67080	0.835399	−	0.549644i	$-0.185237\pi$
$269$	27.4031	1.67080	0.835399	+	0.549644i	$0.185237\pi$
$270$	0	0
$271$	−23.8406	−1.44822	−0.724108	−	0.689686i	$-0.757748\pi$
$271$	−23.8406	−1.44822	−0.724108	+	0.689686i	$0.757748\pi$
$272$	0	0
$273$	−4.70156	−0.284551
$274$	0	0
$275$	1.77572	0.107080
$276$	0	0
$277$	−12.0000	−0.721010	−0.360505	−	0.932757i	$-0.617396\pi$
$277$	−12.0000	−0.721010	−0.360505	+	0.932757i	$0.617396\pi$
$278$	0	0
$279$	−1.77572	−0.106310
$280$	0	0
$281$	−11.4031	−0.680253	−0.340127	−	0.940380i	$-0.610470\pi$
$281$	−11.4031	−0.680253	−0.340127	+	0.940380i	$0.610470\pi$
$282$	0	0
$283$	−1.54515	−0.0918499	−0.0459250	−	0.998945i	$-0.514624\pi$
$283$	−1.54515	−0.0918499	−0.0459250	+	0.998945i	$0.514624\pi$
$284$	0	0
$285$	21.6125	1.28021
$286$	0	0
$287$	29.5197	1.74249
$288$	0	0
$289$	−16.5078	−0.971048
$290$	0	0
$291$	18.1616	1.06465
$292$	0	0
$293$	−8.10469	−0.473481	−0.236740	−	0.971573i	$-0.576079\pi$
$293$	−8.10469	−0.473481	−0.236740	+	0.971573i	$0.576079\pi$
$294$	0	0
$295$	7.72577	0.449812
$296$	0	0
$297$	−3.79063	−0.219955
$298$	0	0
$299$	−3.63232	−0.210063
$300$	0	0
$301$	26.1047	1.50465
$302$	0	0
$303$	−12.3612	−0.710135
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−23.5696	−1.34519	−0.672595	−	0.740011i	$-0.734820\pi$
$307$	−23.5696	−1.34519	−0.672595	+	0.740011i	$0.734820\pi$
$308$	0	0
$309$	−25.4031	−1.44513
$310$	0	0
$311$	23.8002	1.34959	0.674793	−	0.738007i	$-0.264233\pi$
$311$	23.8002	1.34959	0.674793	+	0.738007i	$0.264233\pi$
$312$	0	0
$313$	−6.10469	−0.345057	−0.172529	−	0.985005i	$-0.555194\pi$
$313$	−6.10469	−0.345057	−0.172529	+	0.985005i	$0.555194\pi$
$314$	0	0
$315$	2.08717	0.117598
$316$	0	0
$317$	−6.00000	−0.336994	−0.168497	−	0.985702i	$-0.553891\pi$
$317$	−6.00000	−0.336994	−0.168497	+	0.985702i	$0.553891\pi$
$318$	0	0
$319$	−1.54515	−0.0865121
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−3.09031	−0.171949
$324$	0	0
$325$	−2.29844	−0.127494
$326$	0	0
$327$	−2.35817	−0.130407
$328$	0	0
$329$	−24.1047	−1.32893
$330$	0	0
$331$	−18.3922	−1.01093	−0.505463	−	0.862849i	$-0.668678\pi$
$331$	−18.3922	−1.01093	−0.505463	+	0.862849i	$0.668678\pi$
$332$	0	0
$333$	2.00000	0.109599
$334$	0	0
$335$	−34.2360	−1.87051
$336$	0	0
$337$	27.5078	1.49845	0.749223	−	0.662318i	$-0.230427\pi$
$337$	27.5078	1.49845	0.749223	+	0.662318i	$0.230427\pi$
$338$	0	0
$339$	−2.54830	−0.138405
$340$	0	0
$341$	−4.59688	−0.248935
$342$	0	0
$343$	−18.8937	−1.02017
$344$	0	0
$345$	17.8219	0.959497
$346$	0	0
$347$	−16.3454	−0.877469	−0.438735	−	0.898617i	$-0.644573\pi$
$347$	−16.3454	−0.877469	−0.438735	+	0.898617i	$0.644573\pi$
$348$	0	0
$349$	2.70156	0.144611	0.0723057	−	0.997383i	$-0.476964\pi$
$349$	2.70156	0.144611	0.0723057	+	0.997383i	$0.476964\pi$
$350$	0	0
$351$	4.90647	0.261888
$352$	0	0
$353$	−24.8062	−1.32030	−0.660152	−	0.751132i	$-0.729508\pi$
$353$	−24.8062	−1.32030	−0.660152	+	0.751132i	$0.729508\pi$
$354$	0	0
$355$	−2.81930	−0.149633
$356$	0	0
$357$	−3.29844	−0.174572
$358$	0	0
$359$	−5.95005	−0.314032	−0.157016	−	0.987596i	$-0.550187\pi$
$359$	−5.95005	−0.314032	−0.157016	+	0.987596i	$0.550187\pi$
$360$	0	0
$361$	0.403124	0.0212171
$362$	0	0
$363$	−18.8937	−0.991664
$364$	0	0
$365$	20.0000	1.04685
$366$	0	0
$367$	36.2423	1.89183	0.945917	−	0.324409i	$-0.105165\pi$
$367$	36.2423	1.89183	0.945917	+	0.324409i	$0.105165\pi$
$368$	0	0
$369$	3.40312	0.177160
$370$	0	0
$371$	−3.63232	−0.188581
$372$	0	0
$373$	3.40312	0.176207	0.0881035	−	0.996111i	$-0.471919\pi$
$373$	3.40312	0.176207	0.0881035	+	0.996111i	$0.471919\pi$
$374$	0	0
$375$	−13.2551	−0.684492
$376$	0	0
$377$	2.00000	0.103005
$378$	0	0
$379$	−3.86289	−0.198423	−0.0992116	−	0.995066i	$-0.531632\pi$
$379$	−3.86289	−0.198423	−0.0992116	+	0.995066i	$0.531632\pi$
$380$	0	0
$381$	−19.7906	−1.01390
$382$	0	0
$383$	−16.0340	−0.819299	−0.409649	−	0.912243i	$-0.634349\pi$
$383$	−16.0340	−0.819299	−0.409649	+	0.912243i	$0.634349\pi$
$384$	0	0
$385$	5.40312	0.275369
$386$	0	0
$387$	3.00943	0.152978
$388$	0	0
$389$	19.4031	0.983777	0.491889	−	0.870658i	$-0.336307\pi$
$389$	19.4031	0.983777	0.491889	+	0.870658i	$0.336307\pi$
$390$	0	0
$391$	−2.54830	−0.128873
$392$	0	0
$393$	−8.91093	−0.449497
$394$	0	0
$395$	5.63861	0.283709
$396$	0	0
$397$	12.8062	0.642727	0.321364	−	0.946956i	$-0.395859\pi$
$397$	12.8062	0.642727	0.321364	+	0.946956i	$0.395859\pi$
$398$	0	0
$399$	20.7099	1.03679
$400$	0	0
$401$	30.0000	1.49813	0.749064	−	0.662497i	$-0.230503\pi$
$401$	30.0000	1.49813	0.749064	+	0.662497i	$0.230503\pi$
$402$	0	0
$403$	5.95005	0.296393
$404$	0	0
$405$	−26.4922	−1.31641
$406$	0	0
$407$	5.17748	0.256638
$408$	0	0
$409$	−36.8062	−1.81995	−0.909976	−	0.414661i	$-0.863900\pi$
$409$	−36.8062	−1.81995	−0.909976	+	0.414661i	$0.863900\pi$
$410$	0	0
$411$	25.4262	1.25418
$412$	0	0
$413$	7.40312	0.364284
$414$	0	0
$415$	25.8874	1.27076
$416$	0	0
$417$	−22.1047	−1.08247
$418$	0	0
$419$	19.4358	0.949499	0.474749	−	0.880121i	$-0.342539\pi$
$419$	19.4358	0.949499	0.474749	+	0.880121i	$0.342539\pi$
$420$	0	0
$421$	−4.10469	−0.200050	−0.100025	−	0.994985i	$-0.531892\pi$
$421$	−4.10469	−0.200050	−0.100025	+	0.994985i	$0.531892\pi$
$422$	0	0
$423$	−2.77886	−0.135113
$424$	0	0
$425$	−1.61250	−0.0782176
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−1.40312	−0.0677435
$430$	0	0
$431$	−15.0309	−0.724011	−0.362005	−	0.932176i	$-0.617908\pi$
$431$	−15.0309	−0.724011	−0.362005	+	0.932176i	$0.617908\pi$
$432$	0	0
$433$	22.9109	1.10103	0.550515	−	0.834826i	$-0.314432\pi$
$433$	22.9109	1.10103	0.550515	+	0.834826i	$0.314432\pi$
$434$	0	0
$435$	−9.81294	−0.470494
$436$	0	0
$437$	16.0000	0.765384
$438$	0	0
$439$	−17.6196	−0.840937	−0.420469	−	0.907307i	$-0.638134\pi$
$439$	−17.6196	−0.840937	−0.420469	+	0.907307i	$0.638134\pi$
$440$	0	0
$441$	−0.0890652	−0.00424120
$442$	0	0
$443$	22.5261	1.07025	0.535123	−	0.844774i	$-0.320265\pi$
$443$	22.5261	1.07025	0.535123	+	0.844774i	$0.320265\pi$
$444$	0	0
$445$	30.8062	1.46036
$446$	0	0
$447$	−23.2582	−1.10008
$448$	0	0
$449$	27.6125	1.30311	0.651557	−	0.758600i	$-0.274116\pi$
$449$	27.6125	1.30311	0.651557	+	0.758600i	$0.274116\pi$
$450$	0	0
$451$	8.80980	0.414837
$452$	0	0
$453$	43.2984	2.03434
$454$	0	0
$455$	−6.99364	−0.327867
$456$	0	0
$457$	−11.4031	−0.533416	−0.266708	−	0.963777i	$-0.585936\pi$
$457$	−11.4031	−0.533416	−0.266708	+	0.963777i	$0.585936\pi$
$458$	0	0
$459$	3.44219	0.160668
$460$	0	0
$461$	−10.9109	−0.508173	−0.254086	−	0.967182i	$-0.581775\pi$
$461$	−10.9109	−0.508173	−0.254086	+	0.967182i	$0.581775\pi$
$462$	0	0
$463$	−14.7598	−0.685948	−0.342974	−	0.939345i	$-0.611434\pi$
$463$	−14.7598	−0.685948	−0.342974	+	0.939345i	$0.611434\pi$
$464$	0	0
$465$	−29.1938	−1.35383
$466$	0	0
$467$	−2.62918	−0.121664	−0.0608319	−	0.998148i	$-0.519375\pi$
$467$	−2.62918	−0.121664	−0.0608319	+	0.998148i	$0.519375\pi$
$468$	0	0
$469$	−32.8062	−1.51485
$470$	0	0
$471$	8.34866	0.384686
$472$	0	0
$473$	7.79063	0.358213
$474$	0	0
$475$	10.1244	0.464539
$476$	0	0
$477$	−0.418745	−0.0191730
$478$	0	0
$479$	−18.1212	−0.827977	−0.413989	−	0.910282i	$-0.635865\pi$
$479$	−18.1212	−0.827977	−0.413989	+	0.910282i	$0.635865\pi$
$480$	0	0
$481$	−6.70156	−0.305565
$482$	0	0
$483$	17.0776	0.777057
$484$	0	0
$485$	27.0156	1.22672
$486$	0	0
$487$	12.1307	0.549693	0.274847	−	0.961488i	$-0.411373\pi$
$487$	12.1307	0.549693	0.274847	+	0.961488i	$0.411373\pi$
$488$	0	0
$489$	−30.5969	−1.38364
$490$	0	0
$491$	27.7035	1.25024	0.625122	−	0.780527i	$-0.285049\pi$
$491$	27.7035	1.25024	0.625122	+	0.780527i	$0.285049\pi$
$492$	0	0
$493$	1.40312	0.0631935
$494$	0	0
$495$	0.622889	0.0279968
$496$	0	0
$497$	−2.70156	−0.121182
$498$	0	0
$499$	14.7598	0.660742	0.330371	−	0.943851i	$-0.392826\pi$
$499$	14.7598	0.660742	0.330371	+	0.943851i	$0.392826\pi$
$500$	0	0
$501$	−26.8062	−1.19761
$502$	0	0
$503$	−23.3391	−1.04064	−0.520319	−	0.853972i	$-0.674187\pi$
$503$	−23.3391	−1.04064	−0.520319	+	0.853972i	$0.674187\pi$
$504$	0	0
$505$	−18.3875	−0.818233
$506$	0	0
$507$	1.81616	0.0806585
$508$	0	0
$509$	−35.6125	−1.57850	−0.789248	−	0.614074i	$-0.789529\pi$
$509$	−35.6125	−1.57850	−0.789248	+	0.614074i	$0.789529\pi$
$510$	0	0
$511$	19.1647	0.847798
$512$	0	0
$513$	−21.6125	−0.954215
$514$	0	0
$515$	−37.7875	−1.66512
$516$	0	0
$517$	−7.19375	−0.316381
$518$	0	0
$519$	−38.8715	−1.70627
$520$	0	0
$521$	−6.10469	−0.267451	−0.133726	−	0.991018i	$-0.542694\pi$
$521$	−6.10469	−0.267451	−0.133726	+	0.991018i	$0.542694\pi$
$522$	0	0
$523$	41.4198	1.81116	0.905581	−	0.424174i	$-0.139436\pi$
$523$	41.4198	1.81116	0.905581	+	0.424174i	$0.139436\pi$
$524$	0	0
$525$	10.8062	0.471623
$526$	0	0
$527$	4.17433	0.181837
$528$	0	0
$529$	−9.80625	−0.426359
$530$	0	0
$531$	0.853456	0.0370368
$532$	0	0
$533$	−11.4031	−0.493924
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−43.7172	−1.88654
$538$	0	0
$539$	−0.230566	−0.00993120
$540$	0	0
$541$	25.2984	1.08766	0.543832	−	0.839194i	$-0.316973\pi$
$541$	25.2984	1.08766	0.543832	+	0.839194i	$0.316973\pi$
$542$	0	0
$543$	−25.4262	−1.09114
$544$	0	0
$545$	−3.50781	−0.150258
$546$	0	0
$547$	−12.7131	−0.543574	−0.271787	−	0.962357i	$-0.587615\pi$
$547$	−12.7131	−0.543574	−0.271787	+	0.962357i	$0.587615\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−8.80980	−0.375310
$552$	0	0
$553$	5.40312	0.229764
$554$	0	0
$555$	32.8810	1.39572
$556$	0	0
$557$	−38.7016	−1.63984	−0.819919	−	0.572480i	$-0.805982\pi$
$557$	−38.7016	−1.63984	−0.819919	+	0.572480i	$0.805982\pi$
$558$	0	0
$559$	−10.0839	−0.426505
$560$	0	0
$561$	−0.984379	−0.0415605
$562$	0	0
$563$	−19.8969	−0.838554	−0.419277	−	0.907858i	$-0.637716\pi$
$563$	−19.8969	−0.838554	−0.419277	+	0.907858i	$0.637716\pi$
$564$	0	0
$565$	−3.79063	−0.159473
$566$	0	0
$567$	−25.3858	−1.06610
$568$	0	0
$569$	28.1047	1.17821	0.589105	−	0.808057i	$-0.299480\pi$
$569$	28.1047	1.17821	0.589105	+	0.808057i	$0.299480\pi$
$570$	0	0
$571$	−0.813017	−0.0340237	−0.0170118	−	0.999855i	$-0.505415\pi$
$571$	−0.813017	−0.0340237	−0.0170118	+	0.999855i	$0.505415\pi$
$572$	0	0
$573$	−9.40312	−0.392821
$574$	0	0
$575$	8.34866	0.348163
$576$	0	0
$577$	1.79063	0.0745448	0.0372724	−	0.999305i	$-0.488133\pi$
$577$	1.79063	0.0745448	0.0372724	+	0.999305i	$0.488133\pi$
$578$	0	0
$579$	37.7875	1.57039
$580$	0	0
$581$	24.8062	1.02914
$582$	0	0
$583$	−1.08402	−0.0448956
$584$	0	0
$585$	−0.806248	−0.0333343
$586$	0	0
$587$	30.2923	1.25030	0.625148	−	0.780506i	$-0.285039\pi$
$587$	30.2923	1.25030	0.625148	+	0.780506i	$0.285039\pi$
$588$	0	0
$589$	−26.2094	−1.07994
$590$	0	0
$591$	0.190127	0.00782079
$592$	0	0
$593$	14.0000	0.574911	0.287456	−	0.957794i	$-0.407191\pi$
$593$	14.0000	0.574911	0.287456	+	0.957794i	$0.407191\pi$
$594$	0	0
$595$	−4.90647	−0.201146
$596$	0	0
$597$	−12.2094	−0.499696
$598$	0	0
$599$	25.3454	1.03558	0.517792	−	0.855507i	$-0.326754\pi$
$599$	25.3454	1.03558	0.517792	+	0.855507i	$0.326754\pi$
$600$	0	0
$601$	−34.3141	−1.39970	−0.699850	−	0.714290i	$-0.746750\pi$
$601$	−34.3141	−1.39970	−0.699850	+	0.714290i	$0.746750\pi$
$602$	0	0
$603$	−3.78201	−0.154015
$604$	0	0
$605$	−28.1047	−1.14262
$606$	0	0
$607$	33.6940	1.36760	0.683799	−	0.729670i	$-0.260326\pi$
$607$	33.6940	1.36760	0.683799	+	0.729670i	$0.260326\pi$
$608$	0	0
$609$	−9.40312	−0.381034
$610$	0	0
$611$	9.31137	0.376698
$612$	0	0
$613$	−46.4187	−1.87484	−0.937418	−	0.348207i	$-0.886791\pi$
$613$	−46.4187	−1.87484	−0.937418	+	0.348207i	$0.886791\pi$
$614$	0	0
$615$	55.9491	2.25608
$616$	0	0
$617$	1.79063	0.0720879	0.0360440	−	0.999350i	$-0.488524\pi$
$617$	1.79063	0.0720879	0.0360440	+	0.999350i	$0.488524\pi$
$618$	0	0
$619$	−15.2210	−0.611783	−0.305891	−	0.952066i	$-0.598954\pi$
$619$	−15.2210	−0.611783	−0.305891	+	0.952066i	$0.598954\pi$
$620$	0	0
$621$	−17.8219	−0.715167
$622$	0	0
$623$	29.5197	1.18268
$624$	0	0
$625$	−31.2094	−1.24837
$626$	0	0
$627$	6.18062	0.246830
$628$	0	0
$629$	−4.70156	−0.187464
$630$	0	0
$631$	37.2859	1.48433	0.742164	−	0.670218i	$-0.233799\pi$
$631$	37.2859	1.48433	0.742164	+	0.670218i	$0.233799\pi$
$632$	0	0
$633$	15.5078	0.616380
$634$	0	0
$635$	−29.4388	−1.16824
$636$	0	0
$637$	0.298438	0.0118245
$638$	0	0
$639$	−0.311445	−0.0123206
$640$	0	0
$641$	−8.20937	−0.324251	−0.162125	−	0.986770i	$-0.551835\pi$
$641$	−8.20937	−0.324251	−0.162125	+	0.986770i	$0.551835\pi$
$642$	0	0
$643$	19.9373	0.786251	0.393126	−	0.919485i	$-0.371394\pi$
$643$	19.9373	0.786251	0.393126	+	0.919485i	$0.371394\pi$
$644$	0	0
$645$	49.4766	1.94814
$646$	0	0
$647$	−14.5293	−0.571205	−0.285603	−	0.958348i	$-0.592194\pi$
$647$	−14.5293	−0.571205	−0.285603	+	0.958348i	$0.592194\pi$
$648$	0	0
$649$	2.20937	0.0867255
$650$	0	0
$651$	−27.9745	−1.09641
$652$	0	0
$653$	23.4031	0.915835	0.457918	−	0.888995i	$-0.348596\pi$
$653$	23.4031	0.915835	0.457918	+	0.888995i	$0.348596\pi$
$654$	0	0
$655$	−13.2551	−0.517921
$656$	0	0
$657$	2.20937	0.0861958
$658$	0	0
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	0	0
$661$	−10.0000	−0.388955	−0.194477	−	0.980907i	$-0.562301\pi$
$661$	−10.0000	−0.388955	−0.194477	+	0.980907i	$0.562301\pi$
$662$	0	0
$663$	1.27415	0.0494839
$664$	0	0
$665$	30.8062	1.19462
$666$	0	0
$667$	−7.26464	−0.281288
$668$	0	0
$669$	−36.7016	−1.41896
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−46.3141	−1.78528	−0.892638	−	0.450774i	$-0.851148\pi$
$673$	−46.3141	−1.78528	−0.892638	+	0.450774i	$0.851148\pi$
$674$	0	0
$675$	−11.2772	−0.434060
$676$	0	0
$677$	−15.0156	−0.577097	−0.288549	−	0.957465i	$-0.593173\pi$
$677$	−15.0156	−0.577097	−0.288549	+	0.957465i	$0.593173\pi$
$678$	0	0
$679$	25.8874	0.993466
$680$	0	0
$681$	0.418745	0.0160464
$682$	0	0
$683$	−46.8278	−1.79182	−0.895909	−	0.444238i	$-0.853474\pi$
$683$	−46.8278	−1.79182	−0.895909	+	0.444238i	$0.853474\pi$
$684$	0	0
$685$	37.8219	1.44510
$686$	0	0
$687$	41.6099	1.58752
$688$	0	0
$689$	1.40312	0.0534548
$690$	0	0
$691$	1.31459	0.0500093	0.0250046	−	0.999687i	$-0.492040\pi$
$691$	1.31459	0.0500093	0.0250046	+	0.999687i	$0.492040\pi$
$692$	0	0
$693$	0.596876	0.0226734
$694$	0	0
$695$	−32.8810	−1.24725
$696$	0	0
$697$	−8.00000	−0.303022
$698$	0	0
$699$	39.0616	1.47745
$700$	0	0
$701$	−8.80625	−0.332607	−0.166304	−	0.986075i	$-0.553183\pi$
$701$	−8.80625	−0.332607	−0.166304	+	0.986075i	$0.553183\pi$
$702$	0	0
$703$	29.5197	1.11336
$704$	0	0
$705$	−45.6859	−1.72063
$706$	0	0
$707$	−17.6196	−0.662653
$708$	0	0
$709$	−14.0000	−0.525781	−0.262891	−	0.964826i	$-0.584676\pi$
$709$	−14.0000	−0.525781	−0.262891	+	0.964826i	$0.584676\pi$
$710$	0	0
$711$	0.622889	0.0233602
$712$	0	0
$713$	−21.6125	−0.809394
$714$	0	0
$715$	−2.08717	−0.0780556
$716$	0	0
$717$	50.7328	1.89465
$718$	0	0
$719$	5.63861	0.210285	0.105142	−	0.994457i	$-0.466470\pi$
$719$	5.63861	0.210285	0.105142	+	0.994457i	$0.466470\pi$
$720$	0	0
$721$	−36.2094	−1.34851
$722$	0	0
$723$	27.9745	1.04038
$724$	0	0
$725$	−4.59688	−0.170724
$726$	0	0
$727$	15.0713	0.558963	0.279482	−	0.960151i	$-0.409837\pi$
$727$	15.0713	0.558963	0.279482	+	0.960151i	$0.409837\pi$
$728$	0	0
$729$	23.8062	0.881713
$730$	0	0
$731$	−7.07451	−0.261660
$732$	0	0
$733$	−39.1203	−1.44494	−0.722471	−	0.691401i	$-0.756994\pi$
$733$	−39.1203	−1.44494	−0.722471	+	0.691401i	$0.756994\pi$
$734$	0	0
$735$	−1.46428	−0.0540106
$736$	0	0
$737$	−9.79063	−0.360642
$738$	0	0
$739$	−4.40490	−0.162037	−0.0810184	−	0.996713i	$-0.525817\pi$
$739$	−4.40490	−0.162037	−0.0810184	+	0.996713i	$0.525817\pi$
$740$	0	0
$741$	−8.00000	−0.293887
$742$	0	0
$743$	−21.2115	−0.778173	−0.389087	−	0.921201i	$-0.627209\pi$
$743$	−21.2115	−0.778173	−0.389087	+	0.921201i	$0.627209\pi$
$744$	0	0
$745$	−34.5969	−1.26753
$746$	0	0
$747$	2.85974	0.104633
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−17.6196	−0.642948	−0.321474	−	0.946918i	$-0.604178\pi$
$751$	−17.6196	−0.642948	−0.321474	+	0.946918i	$0.604178\pi$
$752$	0	0
$753$	−48.0000	−1.74922
$754$	0	0
$755$	64.4070	2.34401
$756$	0	0
$757$	−28.2094	−1.02529	−0.512644	−	0.858602i	$-0.671334\pi$
$757$	−28.2094	−1.02529	−0.512644	+	0.858602i	$0.671334\pi$
$758$	0	0
$759$	5.09660	0.184995
$760$	0	0
$761$	−11.4031	−0.413363	−0.206681	−	0.978408i	$-0.566266\pi$
$761$	−11.4031	−0.413363	−0.206681	+	0.978408i	$0.566266\pi$
$762$	0	0
$763$	−3.36131	−0.121688
$764$	0	0
$765$	−0.565633	−0.0204505
$766$	0	0
$767$	−2.85974	−0.103259
$768$	0	0
$769$	−27.4031	−0.988182	−0.494091	−	0.869410i	$-0.664499\pi$
$769$	−27.4031	−0.988182	−0.494091	+	0.869410i	$0.664499\pi$
$770$	0	0
$771$	19.4358	0.699961
$772$	0	0
$773$	42.7016	1.53587	0.767934	−	0.640529i	$-0.221285\pi$
$773$	42.7016	1.53587	0.767934	+	0.640529i	$0.221285\pi$
$774$	0	0
$775$	−13.6758	−0.491250
$776$	0	0
$777$	31.5078	1.13034
$778$	0	0
$779$	50.2296	1.79966
$780$	0	0
$781$	−0.806248	−0.0288498
$782$	0	0
$783$	9.81294	0.350686
$784$	0	0
$785$	12.4187	0.443244
$786$	0	0
$787$	−26.6600	−0.950325	−0.475162	−	0.879898i	$-0.657611\pi$
$787$	−26.6600	−0.950325	−0.475162	+	0.879898i	$0.657611\pi$
$788$	0	0
$789$	34.8062	1.23914
$790$	0	0
$791$	−3.63232	−0.129150
$792$	0	0
$793$	0	0
$794$	0	0
$795$	−6.88439	−0.244164
$796$	0	0
$797$	−46.2094	−1.63682	−0.818410	−	0.574635i	$-0.805144\pi$
$797$	−46.2094	−1.63682	−0.818410	+	0.574635i	$0.805144\pi$
$798$	0	0
$799$	6.53250	0.231103
$800$	0	0
$801$	3.40312	0.120243
$802$	0	0
$803$	5.71949	0.201836
$804$	0	0
$805$	25.4031	0.895342
$806$	0	0
$807$	49.7685	1.75193
$808$	0	0
$809$	31.5078	1.10776	0.553878	−	0.832598i	$-0.313147\pi$
$809$	31.5078	1.10776	0.553878	+	0.832598i	$0.313147\pi$
$810$	0	0
$811$	−38.0989	−1.33783	−0.668917	−	0.743337i	$-0.733242\pi$
$811$	−38.0989	−1.33783	−0.668917	+	0.743337i	$0.733242\pi$
$812$	0	0
$813$	−43.2984	−1.51854
$814$	0	0
$815$	−45.5133	−1.59426
$816$	0	0
$817$	44.4187	1.55402
$818$	0	0
$819$	−0.772577	−0.0269960
$820$	0	0
$821$	−18.4922	−0.645382	−0.322691	−	0.946504i	$-0.604587\pi$
$821$	−18.4922	−0.645382	−0.322691	+	0.946504i	$0.604587\pi$
$822$	0	0
$823$	−34.1552	−1.19057	−0.595287	−	0.803513i	$-0.702962\pi$
$823$	−34.1552	−1.19057	−0.595287	+	0.803513i	$0.702962\pi$
$824$	0	0
$825$	3.22499	0.112280
$826$	0	0
$827$	2.85974	0.0994430	0.0497215	−	0.998763i	$-0.484167\pi$
$827$	2.85974	0.0994430	0.0497215	+	0.998763i	$0.484167\pi$
$828$	0	0
$829$	37.4031	1.29906	0.649532	−	0.760334i	$-0.274965\pi$
$829$	37.4031	1.29906	0.649532	+	0.760334i	$0.274965\pi$
$830$	0	0
$831$	−21.7939	−0.756023
$832$	0	0
$833$	0.209373	0.00725433
$834$	0	0
$835$	−39.8746	−1.37992
$836$	0	0
$837$	29.1938	1.00908
$838$	0	0
$839$	53.0893	1.83285	0.916424	−	0.400209i	$-0.131063\pi$
$839$	53.0893	1.83285	0.916424	+	0.400209i	$0.131063\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	−20.7099	−0.713287
$844$	0	0
$845$	2.70156	0.0929366
$846$	0	0
$847$	−26.9310	−0.925359
$848$	0	0
$849$	−2.80625	−0.0963102
$850$	0	0
$851$	24.3422	0.834441
$852$	0	0
$853$	21.2984	0.729245	0.364622	−	0.931156i	$-0.381198\pi$
$853$	21.2984	0.729245	0.364622	+	0.931156i	$0.381198\pi$
$854$	0	0
$855$	3.55144	0.121457
$856$	0	0
$857$	50.4187	1.72227	0.861136	−	0.508375i	$-0.169754\pi$
$857$	50.4187	1.72227	0.861136	+	0.508375i	$0.169754\pi$
$858$	0	0
$859$	38.3295	1.30779	0.653893	−	0.756587i	$-0.273135\pi$
$859$	38.3295	1.30779	0.653893	+	0.756587i	$0.273135\pi$
$860$	0	0
$861$	53.6125	1.82711
$862$	0	0
$863$	−21.2115	−0.722047	−0.361023	−	0.932557i	$-0.617573\pi$
$863$	−21.2115	−0.722047	−0.361023	+	0.932557i	$0.617573\pi$
$864$	0	0
$865$	−57.8219	−1.96600
$866$	0	0
$867$	−29.9808	−1.01820
$868$	0	0
$869$	1.61250	0.0547002
$870$	0	0
$871$	12.6727	0.429397
$872$	0	0
$873$	2.98438	0.101006
$874$	0	0
$875$	−18.8937	−0.638725
$876$	0	0
$877$	−48.3141	−1.63145	−0.815725	−	0.578440i	$-0.803662\pi$
$877$	−48.3141	−1.63145	−0.815725	+	0.578440i	$0.803662\pi$
$878$	0	0
$879$	−14.7194	−0.496473
$880$	0	0
$881$	−38.9109	−1.31094	−0.655471	−	0.755220i	$-0.727530\pi$
$881$	−38.9109	−1.31094	−0.655471	+	0.755220i	$0.727530\pi$
$882$	0	0
$883$	−11.1680	−0.375832	−0.187916	−	0.982185i	$-0.560173\pi$
$883$	−11.1680	−0.375832	−0.187916	+	0.982185i	$0.560173\pi$
$884$	0	0
$885$	14.0312	0.471655
$886$	0	0
$887$	−42.9650	−1.44262	−0.721311	−	0.692611i	$-0.756460\pi$
$887$	−42.9650	−1.44262	−0.721311	+	0.692611i	$0.756460\pi$
$888$	0	0
$889$	−28.2094	−0.946112
$890$	0	0
$891$	−7.57609	−0.253808
$892$	0	0
$893$	−41.0156	−1.37254
$894$	0	0
$895$	−65.0299	−2.17371
$896$	0	0
$897$	−6.59688	−0.220263
$898$	0	0
$899$	11.9001	0.396891
$900$	0	0
$901$	0.984379	0.0327944
$902$	0	0
$903$	47.4103	1.57772
$904$	0	0
$905$	−37.8219	−1.25724
$906$	0	0
$907$	−41.6908	−1.38432	−0.692160	−	0.721744i	$-0.743341\pi$
$907$	−41.6908	−1.38432	−0.692160	+	0.721744i	$0.743341\pi$
$908$	0	0
$909$	−2.03124	−0.0673721
$910$	0	0
$911$	−19.7068	−0.652914	−0.326457	−	0.945212i	$-0.605855\pi$
$911$	−19.7068	−0.652914	−0.326457	+	0.945212i	$0.605855\pi$
$912$	0	0
$913$	7.40312	0.245008
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−12.7016	−0.419443
$918$	0	0
$919$	−28.4357	−0.938006	−0.469003	−	0.883196i	$-0.655387\pi$
$919$	−28.4357	−0.938006	−0.469003	+	0.883196i	$0.655387\pi$
$920$	0	0
$921$	−42.8062	−1.41051
$922$	0	0
$923$	1.04358	0.0343499
$924$	0	0
$925$	15.4031	0.506452
$926$	0	0
$927$	−4.17433	−0.137103
$928$	0	0
$929$	20.5969	0.675762	0.337881	−	0.941189i	$-0.390290\pi$
$929$	20.5969	0.675762	0.337881	+	0.941189i	$0.390290\pi$
$930$	0	0
$931$	−1.31459	−0.0430839
$932$	0	0
$933$	43.2250	1.41512
$934$	0	0
$935$	−1.46428	−0.0478870
$936$	0	0
$937$	3.61250	0.118015	0.0590076	−	0.998258i	$-0.481206\pi$
$937$	3.61250	0.118015	0.0590076	+	0.998258i	$0.481206\pi$
$938$	0	0
$939$	−11.0871	−0.361813
$940$	0	0
$941$	52.5234	1.71221	0.856107	−	0.516798i	$-0.172876\pi$
$941$	52.5234	1.71221	0.856107	+	0.516798i	$0.172876\pi$
$942$	0	0
$943$	41.4198	1.34881
$944$	0	0
$945$	−34.3141	−1.11624
$946$	0	0
$947$	−37.0958	−1.20545	−0.602725	−	0.797949i	$-0.705919\pi$
$947$	−37.0958	−1.20545	−0.602725	+	0.797949i	$0.705919\pi$
$948$	0	0
$949$	−7.40312	−0.240316
$950$	0	0
$951$	−10.8970	−0.353358
$952$	0	0
$953$	13.2984	0.430779	0.215389	−	0.976528i	$-0.430898\pi$
$953$	13.2984	0.430779	0.215389	+	0.976528i	$0.430898\pi$
$954$	0	0
$955$	−13.9873	−0.452617
$956$	0	0
$957$	−2.80625	−0.0907131
$958$	0	0
$959$	36.2423	1.17033
$960$	0	0
$961$	4.40312	0.142036
$962$	0	0
$963$	0	0
$964$	0	0
$965$	56.2094	1.80944
$966$	0	0
$967$	2.04673	0.0658183	0.0329091	−	0.999458i	$-0.489523\pi$
$967$	2.04673	0.0658183	0.0329091	+	0.999458i	$0.489523\pi$
$968$	0	0
$969$	−5.61250	−0.180299
$970$	0	0
$971$	1.81616	0.0582834	0.0291417	−	0.999575i	$-0.490723\pi$
$971$	1.81616	0.0582834	0.0291417	+	0.999575i	$0.490723\pi$
$972$	0	0
$973$	−31.5078	−1.01009
$974$	0	0
$975$	−4.17433	−0.133686
$976$	0	0
$977$	7.19375	0.230149	0.115074	−	0.993357i	$-0.463289\pi$
$977$	7.19375	0.230149	0.115074	+	0.993357i	$0.463289\pi$
$978$	0	0
$979$	8.80980	0.281562
$980$	0	0
$981$	−0.387503	−0.0123720
$982$	0	0
$983$	3.04987	0.0972758	0.0486379	−	0.998816i	$-0.484512\pi$
$983$	3.04987	0.0972758	0.0486379	+	0.998816i	$0.484512\pi$
$984$	0	0
$985$	0.282817	0.00901129
$986$	0	0
$987$	−43.7780	−1.39347
$988$	0	0
$989$	36.6281	1.16471
$990$	0	0
$991$	40.9587	1.30109	0.650547	−	0.759466i	$-0.274539\pi$
$991$	40.9587	1.30109	0.650547	+	0.759466i	$0.274539\pi$
$992$	0	0
$993$	−33.4031	−1.06002
$994$	0	0
$995$	−18.1616	−0.575761
$996$	0	0
$997$	12.2094	0.386675	0.193337	−	0.981132i	$-0.438069\pi$
$997$	12.2094	0.386675	0.193337	+	0.981132i	$0.438069\pi$
$998$	0	0
$999$	−32.8810	−1.04031

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3328.2.a.bk.1.3		4
4.3	odd	2	inner	3328.2.a.bk.1.2		4
8.3	odd	2		3328.2.a.bl.1.3		4
8.5	even	2		3328.2.a.bl.1.2		4
16.3	odd	4		1664.2.b.j.833.3	✓	8
16.5	even	4		1664.2.b.j.833.4	yes	8
16.11	odd	4		1664.2.b.j.833.6	yes	8
16.13	even	4		1664.2.b.j.833.5	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1664.2.b.j.833.3	✓	8	16.3	odd	4
1664.2.b.j.833.4	yes	8	16.5	even	4
1664.2.b.j.833.5	yes	8	16.13	even	4
1664.2.b.j.833.6	yes	8	16.11	odd	4
3328.2.a.bk.1.2		4	4.3	odd	2	inner
3328.2.a.bk.1.3		4	1.1	even	1	trivial
3328.2.a.bl.1.2		4	8.5	even	2
3328.2.a.bl.1.3		4	8.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 3328 = 2^{8} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3328.a (trivial)

\(f(q)\)	\(=\)	\(q+1.81616 q^{3} +2.70156 q^{5} +2.58874 q^{7} +0.298438 q^{9} +0.772577 q^{11} -1.00000 q^{13} +4.90647 q^{15} -0.701562 q^{17} +4.40490 q^{19} +4.70156 q^{21} +3.63232 q^{23} +2.29844 q^{25} -4.90647 q^{27} -2.00000 q^{29} -5.95005 q^{31} +1.40312 q^{33} +6.99364 q^{35} +6.70156 q^{37} -1.81616 q^{39} +11.4031 q^{41} +10.0839 q^{43} +0.806248 q^{45} -9.31137 q^{47} -0.298438 q^{49} -1.27415 q^{51} -1.40312 q^{53} +2.08717 q^{55} +8.00000 q^{57} +2.85974 q^{59} +0.772577 q^{63} -2.70156 q^{65} -12.6727 q^{67} +6.59688 q^{69} -1.04358 q^{71} +7.40312 q^{73} +4.17433 q^{75} +2.00000 q^{77} +2.08717 q^{79} -9.80625 q^{81} +9.58237 q^{83} -1.89531 q^{85} -3.63232 q^{87} +11.4031 q^{89} -2.58874 q^{91} -10.8062 q^{93} +11.9001 q^{95} +10.0000 q^{97} +0.230566 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{5} + 14 q^{9} - 4 q^{13} + 10 q^{17} + 6 q^{21} + 22 q^{25} - 8 q^{29} - 20 q^{33} + 14 q^{37} + 20 q^{41} - 48 q^{45} - 14 q^{49} + 20 q^{53} + 32 q^{57} + 2 q^{65} + 52 q^{69} + 4 q^{73} + 8 q^{77}+ \cdots + 40 q^{97}+O(q^{100}) \) 4 * q - 2 * q^5 + 14 * q^9 - 4 * q^13 + 10 * q^17 + 6 * q^21 + 22 * q^25 - 8 * q^29 - 20 * q^33 + 14 * q^37 + 20 * q^41 - 48 * q^45 - 14 * q^49 + 20 * q^53 + 32 * q^57 + 2 * q^65 + 52 * q^69 + 4 * q^73 + 8 * q^77 + 12 * q^81 - 46 * q^85 + 20 * q^89 + 8 * q^93 + 40 * q^97

Introduction

L-functions

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